Lyapunov Characterization for ISS of Impulsive Switched Systems

In this study, we investigate the ISS of impulsive switched systems that have modes with both stable and unstable flows. We assume that the switching signal satisfies mode-dependent average dwell and leave time conditions. To establish ISS conditions, we propose two types of time-varying ISS-Lyapunov functions: one that is non-decreasing and another one that is decreasing. Our research proves that the existence of either of these ISS-Lyapunov functions is a necessary and sufficient condition for ISS. We also present a technique for constructing a decreasing ISS-Lyapunov function from a non-decreasing one, which is useful for its own sake. Our findings also have added value to previous research that only studied sufficient conditions for ISS, as our results apply to a broader class of systems. This is because we impose less restrictive dwell and leave time constraints on the switching signal and our ISS-Lyapunov functions are time-varying with general nonlinear conditions imposed on them. Moreover, we provide a method to guarantee the ISS of a particular class of impulsive switched systems when the switching signal is unknown.

💡 Research Summary

The paper addresses the input‑to‑state stability (ISS) of hybrid systems that combine impulsive dynamics with mode‑switching. The authors consider a broad class of systems in which each “mode’’ consists of a continuous flow followed by an instantaneous jump, and they allow the set of modes to be finite or infinite. Crucially, some modes have stable flows while others have unstable flows. To capture this heterogeneity, the switching signal is required to satisfy two mode‑dependent timing constraints: (i) a mode‑dependent average dwell‑time (MD‑ADT) for the set of stable‑flow modes, and (ii) a mode‑dependent average leave‑time (MD‑ALT) for the set of unstable‑flow modes. The MD‑ADT condition limits the number of switches relative to the total time spent in stable modes, ensuring that stable dynamics dominate; the MD‑ALT condition imposes a lower bound on the frequency of switches in unstable modes, guaranteeing that frequent jumps can counteract the instability of the flow.

The core theoretical contribution is the introduction of two families of time‑varying ISS‑Lyapunov functions. A non‑decreasing ISS‑Lyapunov function is allowed to increase during flow intervals but must satisfy a differential inequality of the form
(\dot V(t,x) \le \varphi_{\sigma(t)}(V(t,x)))
whenever (V) exceeds a gain function of the input magnitude. At switching instants it must obey a multiplicative jump inequality. The existence of such a function provides a sufficient condition for ISS. A decreasing ISS‑Lyapunov function, by contrast, is required to satisfy a strict decay inequality during flows and a non‑expansive condition at jumps; its existence is shown to be a necessary condition for ISS. By proving that the two conditions are equivalent—i.e., a system is ISS if and only if it admits both a non‑decreasing and a decreasing ISS‑Lyapunov function—the authors establish a full converse Lyapunov theorem for impulsive switched systems, which had not been available before.

A notable methodological advance is a constructive procedure that transforms any non‑decreasing ISS‑Lyapunov function into a decreasing one. The transformation relies on integrating the absolute value of the decay functions (\varphi_p) for each mode to obtain strictly increasing functions (\Phi_p) and their inverses. By composing the original Lyapunov candidate with (\Phi^{-1}), the authors obtain a new candidate that inherits the level‑set structure of the original but now satisfies a genuine decay condition during flows. This bridge between the two Lyapunov classes clarifies the relationship between sufficient and necessary ISS conditions and provides a practical tool for analysis.

The paper also treats the case where the switching signal is unknown. For linear systems with time‑independent flow and jump maps, the authors specialize the Lyapunov candidates to quadratic forms and derive linear matrix inequality (LMI) conditions that guarantee ISS for all switching signals satisfying the MD‑ADT/MD‑ALT constraints. These LMIs are constructive: feasibility yields a common quadratic Lyapunov matrix (P) and explicit bounds on the input gain, thus enabling controller synthesis or verification without knowledge of the exact switching schedule.

Compared with prior work, the contributions are threefold: (1) the mode‑dependent dwell/leave time constraints are less restrictive than classical dwell‑time conditions and allow an infinite number of modes; (2) the use of time‑varying Lyapunov functions extends ISS analysis to systems where both flows and jumps may be simultaneously unstable, a scenario not covered by earlier sufficient‑only results; (3) the converse Lyapunov theorem and the constructive conversion from non‑decreasing to decreasing functions provide a complete characterization of ISS, closing a gap in the literature.

The paper is organized as follows: Section II introduces notation, the hybrid system model, and the precise definitions of MD‑ADT and MD‑ALT. Section III presents the sufficient ISS condition via a non‑decreasing Lyapunov function. Section IV establishes the necessary and sufficient condition using a decreasing Lyapunov function. Section V details the conversion method between the two Lyapunov classes. Section VI addresses robustness to unknown switching, derives the LMI conditions for linear systems, and illustrates the results with examples. Section VII concludes and outlines future research directions.

In summary, the authors deliver a rigorous, unified framework for ISS of impulsive switched systems, offering both theoretical insight (a converse Lyapunov theorem) and practical tools (constructive Lyapunov conversion and LMI‑based verification) that broaden the applicability of ISS analysis to a wide range of hybrid dynamical systems.